Evidence for Alternative Hypotheses

Most researchers want evidence for the direction of an effect, not evidence against a point null hypothesis. Such evidence is ideally on a scale that is easily interpretable, with an accompanying standard error. Further, the evidence from identical experiments should be repeatable, and evidence from independent experiments should be easily combined, such as required in meta-analysis. Such a measure of evidence exists and has been shown to be closely related to the Kullback-Leibler symmetrized distance between null and alternative hypotheses for exponential families. Here we provide more examples of the latter phenomenon, for distributions lying outside the class of exponential families, including the non-central chi-squared family with unknown non-centrality parameter.